Stress at the Centre of a Square Plate with Linear Boundary Tractions

A unit square homogeneous and isotropic steel plate is centred at the origin of the XY-plane with edges parallel with the coordinate axes and is loaded with linearly distributed normal and tangential boundary tractions as shown in figure 1. The plate can be assumed to be thin so that a plane-stress constitutive relationship is appropriate and for convenience a unit thickness may be used.

Raison d’être for the Challenge

This challenge derives from a philosophical question; can a problem be specified where the response is null? At first sight it seems a little improbable that such a problem can be conceived. However, when one realises that the boundary tractions are applied to the model in the form of consistent nodal forces and that if suitable tractions are chosen such that the consistent nodal forces cancel out then such a problem is easily found. This is the case for the challenge problem when a single four-noded element is used. Further consideration of the problem shows that it possesses a theoretically exact solution which involves linear stress fields that can be captured exactly with a single eight-noded element. The theoretically exact von Mises stress at the centre of the plate is zero and therefore both the single four-noded element and the single eight-noded element predict this value correctly. Disappointingly, however, it will be seen that even though the exact solution is recovered for the single eight-noded element, the available postprocessing facilities in many commercial finite element systems will not allow the user to appreciate this fact because they use linear-interpolation of nodal stress values to simplify plotting procedures.

The solution? Download below, and leave your comments at the bottom of this page!